


Model Selection

by Bitenomnom



Series: Mathematical Proof [14]
Category: Sherlock (TV)
Genre: Drug Use, Gen, Mathematics, Mycroft isn't so bad, Sherlock becomes a Consulting Detective, Sherlock starts working with Lestrade, pre-ASiP, you can put on your Mystrade goggles if you'd like
Language: English
Status: Completed
Published: 2012-09-27
Updated: 2012-09-27
Packaged: 2017-11-15 03:46:28
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,410
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/522789
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p><i>Couldn’t have a promotion with out </i>something<i> going wrong, Greg thought. He would give anything, </i>anything<i> (even his wife, but he already gave her up, didn’t he? Stupid, stupid), to be able to go home and not think about this case, but that wouldn’t happen.</i></p>
<p>Mycroft seeks out newly promoted DI Greg Lestrade in the hopes of finding some way to get Sherlock off his drug habit.</p>
            </blockquote>





	Model Selection

**Author's Note:**

> As the tags would indicate, you can put on Mystrade goggles if you'd like for this one, but it's by no means necessary.
> 
> I feel like I'm probably the only one who'll see where the math comes through in this, but, ah well.

            When attempting to select a model, there are a number of different approaches to help narrow down your options. Among these are the backwards selection method (in which you start with a complete model and remove insignificant variables), the forwards selection method (in which you start with the smallest possible model and add variables), and the stepwise or forwards and backwards method (the forwards method, but with the ability to go backwards if the variables become insignificant over the course of the procedure).

We may also select model candidates based on how well they describe the data using measurements such as Mallows Cp (which depends on the residual sum of squares of the model, of the total model, on the data set size, and on the number of variables), AIC (Akaike’s Information Criteria; depends on the residual sum of squares, the data set size, and the number of parameters in the model), and the BIC (Bayesian Information Criteria; based on the same parameters as AIC). These are similar to the [adjusted R squared](http://archiveofourown.org/works/509493) value, where there is a penalty for having a lot of parameters.

            Having more parameters will always guarantee you a model that better fits your specific points, but at the cost of how well your model will be able to predict anything. You need a model that is adaptable and strikes a balance between describing your data and being able to be projected to other data points that are not a part of the set.

 

***

 

            Greg Lestrade buried his head in his hands.

            Couldn’t have a promotion with out _something_ going wrong, he thought. That something happened to be his team, which was really not favorable for making him look like he knew what he was doing, like he deserved his newly earned position as Detective Inspector.

            The second he was off-duty, he threw his coat around his shoulders and left the office. He carried the case file with him—just in case a revelation came to him this evening, and he needed to check up on any of it. That was, of course, half the problem.

            He would give anything, _anything_ (even his wife, but he already gave her up, didn’t he? Stupid, stupid), to be able to go home and not think about this case, but that wouldn’t happen. It would creep along the edges of his consciousness while he sat down to watch some telly, and then there’d be an advertisement for some crap cop show, and there he’d be again, on the scene, dead man, gunshots from nowhere, holding a knife, no witnesses, no motivation, and it’d be another one to go cold if he couldn’t find something, fast. He’d sit there thinking about it, forget about his beer, forget about sleeping, accidentally doze off in the armchair and wake up four hours later for no reason other than that his body was used to five a.m. It wasn’t speculation: it was exactly what had happened last night.

            “Mycroft, you know this is pointless,” a voice, deep and slightly slurred, echoed down the hallway as Greg exited the lift for the ground floor. All that followed was a sigh from someone else. Greg turned the corner and almost ran into the two men who had positioned themselves not far from the entry. Or perhaps not positioned: One, clearly digging his heels in, the other, trying to lead him further in without looking too much like he was trying to lead him further in.

            “Ah,” said one—different voice than the last one Greg heard, so this must be Mycroft, “Detective Inspector Gregory Lestrade.”

            “I’m off for the day,” he answered, because like bloody hell was he going to personally take reports for another case with this one not even done, still burning against his mind. He could only assume this bloke knew his name because someone up front had told him who to look for. The other man laughed as Greg made to pass the two of them, leave them to bother some other officer, and Greg paused, glancing him over. “What’s he on?”

            “You see how obvious it is?” Mycroft (if that was indeed his name) turned to the other, younger man, whose dark hair was an absolute mess and whose eyes were ringed with what probably amounted to a dangerous lack of sleep. The young man rolled his eyes, and Mycroft took his moment of inattention to stare piercingly at Greg. Greg could wait a moment longer, see what was—well. Maybe this younger fellow needed to be taken into custody.

            “What, so you’re going to have me arrested? Best of luck, that’ll brighten up conversation at the office, won’t it? Make you so much more electable, your imprisoned delinquent little brother.”

            “Don’t be _daft_ , Sherlock,” Mycroft said, but he was searching Greg’s eyes for a solution.

            “What? Slap me with a fine you’ll wind up paying anyway?” the other—Sherlock—answered.

            “Is there something I can do?” Greg cut in, not keen to see any more of the torture but, as per usual with any of his work, drawn in and kept there by the stupid compulsion to do the right thing that he’d been stuck with.

            “I apologize; I never introduced myself.” Mycroft reached out a gloved hand, which Greg grasped firmly. “Mycroft Holmes.”

            “Well, Mister Holmes—”

            “Mycroft,” the man insisted with the sort of force that blew the air out of Greg’s lungs, pronounced with such care that Greg wondered if he was supposed to be honored to be allowed to call this man by his first name. The odd thing was, he _was_.

            “Greg, then,” he responded in kind.

            “This is my brother, Sherlock. As you can see, he has a bit of a,” Mycroft paused, his face contorting, “habit.” The word fell from his tongue like a piece of rubbish no one wanted to touch.

            “I noticed.” Greg meant to focus on Sherlock, his state, what drugs he could be using, but instead his eyes were stuck against Mycroft’s. Sherlock laughed an unpleasant laugh.

            “Unfortunately, incarceration won’t do, and none of the rehabilitation centers will keep him.”  
            “No?”

            “He is…” Mycroft started, and paused, looking to Sherlock again. “Well. Some members of our family have directed our intellect more effectively than he has.”

            This time, Greg did manage to lift his eyes to Sherlock’s pale, sickly skin and dulled eyes. He was—desperate. Not with desire to stop using, not for another fix, just—

            “He hasn’t found an effective outlet for his energy,” Mycroft continued. “He gets…”

            “ _Bored_ ,” Sherlock butted in, clearly not satisfied with whatever list of words Mycroft was thinking through.

            “…Yes. But he has historically proven to be reasonably skilled at solving various problems for interested parties, along the vein of ‘detective work,’ he’d say. You’re a good man,” Mycroft said, with such conviction, as if it were purely a fact, not a compliment, ingratiatingly, coercively. “You could keep an eye on him. Give him something to do.”

            “I really can’t.” The Yard would _certainly_ look down on _that_.

            “Consider it,” Mycroft said. By his tone of voice, the matter was already decided. With careful fingers he retrieved a card from an inside suit pocket and handed it to Greg. “Here is my phone number. You may call at any time; the message will reach me.”

            “Right,” Greg said, because that was all he could think to say in reply. When he tucked the card into his coat pocket, Mycroft gave a brief smile—the fakest smile Greg had seen since the last time they interviewed a serial killer.

When Sherlock turned and all but barreled out the door, Mycroft took in a deep breath. “I hope you can help him. I certainly can’t.” And he turned on his toes and followed Sherlock out.

 

 

 

            It wasn’t going away.

            Greg didn’t even bother turning on the telly this evening, hoping to avoid last night’s troubles, but all that meant was that he could hear everything outside and in the flats around him, cars stopping suddenly and talking and _shouting_. It was nine p.m. and his brain was going nowhere but in circles. Now it had two paths to traverse: first, the murder case; then, Mycroft and his addict brother. It looped from one to the other, pausing on details, blood and bullet holes and Sherlock’s voice saying, “ _Bored,_ ” and Mycroft’s slight shift forward, his guarded eyes not pleading, but almost.

            An evening walk it was, then.

            Greg donned his coat again, unconsciously reaching into his pocket to make sure the card was still there, locked up, and headed out for a brisk walk. It wasn’t much better, but at least it was moving, and perhaps it would wear him out enough physically that he could collapse into bed when he got home. He stopped counting how many streets he’d walked and was debating whether he ought to turn around or go a few more.

            “If you let me see the case files, I could solve it for you in ten minutes,” a deep voice spoke from behind him.

            “Sherlock?” At least it wasn’t a forgettable name.

            “That’s what’s bothering you, isn’t it? That murder you haven’t solved.”

            “How’d you know?”

            Sherlock’s hand waved dismissively. “I’m here to offer my services.”

            “Is this about—”

            “This is not about my brother. It was _my_ idea,” he insisted, and Greg was reminded of his six-year-old son, who he hadn’t seen for—god. Well, maybe he was seven, now. “You don’t call him, you don’t take his offer, whatever exactly he was going to offer you. You let me help you with cases, and perhaps it will distract me enough that I won’t get so _unbearably_ bored.”

            Greg bit his lip. “Look, I’d really like to help, but I can’t exactly take random blokes off the street onto cases—”

            “Consulting detective,” Sherlock said, blurting it as if it had just come straight to his mind and been sent the express route to his vocal cords.

            “What’s that?”

            “That could be—that’s my title. Consulting detective. Not _some random bloke off the street_.”

            Greg considered it. It certainly sounded legitimate enough… “How do I know you’d be helpful?”

            “Give me ten minutes and the case files. I’ll prove it.”

            “And if I let you in on some cases, you’ll stop using?”

            “I don’t _want_ to shoot up, do you see? I _have_ to. So if I don’t have to—well. Problem solved.” Greg arched an eyebrow. He’d have to see about that—find out where Sherlock lived, check up on him now and again to make sure he was holding up his end of the bargain (if that’s what this was; Greg supposed Sherlock had never _agreed_ to anything).

            “You don’t breathe a word of this to anyone,” Greg said, turning back and motioning for Sherlock to follow him. “ _Including_ friends, no matter how trustworthy you might think they are.”

            “I don’t have friends,” Sherlock said. Greg gave him a long glance over his shoulder.

            “The life of a genius is a lonely one, I take it?” he tried. Sherlock said nothing.

 

 

 

            When they got to Greg’s flat, he realized what a state Sherlock was in. For how well-groomed Mycroft had appeared, Sherlock was messy, scruffy, his clothes rumpled and his hair even more so. He was tall, to be sure, with striking features, but now, occupying Greg’s chair, Sherlock was small, hunched, folded in on himself, nothing but a pile of limbs. Greg wondered how lonely a man could be, and then solemnly remembered that he hadn’t gone out for a pint with someone in well over a month, spent all his time at work or alone at home or somewhere traveling in-between. But then, neither Mycroft nor Sherlock seemed like _drinking buddy_ material. He handed Sherlock the files and hovered nearby for a few minutes before asking, “Got anything?”

            “Seven theories,” Sherlock said. “None of them, I daresay, anything that you’ve considered.”

            Greg raised his eyebrows. “Oh? Let’s hear ‘em.”

            “You don’t want me to narrow it down further first?”

            “Oh—well, sure.” He took a seat on his sofa for several minutes more.

            “If the upstairs neighbor recently painted her kitchen, the murderer was an amateur hunter desperate for money,” he finally said.

            “God, you’re making this up!” Greg was about to snatch the files back when Sherlock stood up, effectively putting himself in Greg’s face.

            “Let me explain,” he said, and he did, and Greg had to sit back down or risk falling over with a combination of awe and relief as Sherlock walked him through his conclusions. By the end, Sherlock was grinning, which was a completely inappropriate thing to do in relation to a murder, but Greg thought perhaps he was a bit justified.

            “If that’s what you can do coming off a high, I’d shudder to see what you can do sober.” He paused. “Well, by which I mean, I’d like to see it.”

            “You’ll give me more cases, then?”

            “The really tricky ones, when we can’t figure them out, sure.”

            “Tell your team they’re idiots.”

            “Rather keep my job,” Greg said.

            “You clearly aren’t fit to solve these all on your own, which is what you appear to be trying to do. Meanwhile, they pick through every little detail in the hopes that putting them all together will result in something meaningful. Tell me, they catalogued the texture of the knife he was holding, didn’t they?”

            “Yeah.”

            “Meaningless. Some details merely waste time. _I_ know which. I can point them out when you inevitably miss them.”

            And maybe this was the best way, then, god help him. A little cocksureness was worth putting up with, if what Sherlock said was true.

            “You’re going to call my brother anyway,” Sherlock added after several moments of silence, some of the spark of his excitement gone.

            “I expect so.”

            “Don’t make it about this,” he said it with force, doing his own pleading. Ah, Greg thought: little brother wants to be his own man, tired of his older brother trying to fix his life for him. In this case, though, he should be grateful.

            “What should I make it about, then?”

            “Whatever else you please,” Sherlock said, and Greg got the feeling Sherlock knew something he didn’t. He got the feeling he’d be getting that feeling a lot from now on. The best he could do was hope it’d be worth it.


End file.
